STAT 2655 Assignment 1 - Due in class October 7, 2015 1) Let S contain four sample points E1 , E2 , E3 and E4 . (a) List all possible events in S (include the empty set). (b) Let A and B be the events E1 , E2 , E3 and E2 , E4 respectively. Give the sample points in the following events: A B, A B, Ac B c and Ac B. 2) Five cards are drawn from a standard deck. What is the probability of getting one pair and at least two hearts? 3) (a) A box contains n = 5 dierent and distinguishable pairs of gloves. In a dark room a person randomly selects four gloves. What is the probability that there is no matching pair in the sample? (b) A box contains n dierent and distinguishable pairs of gloves. In a dark room a person randomly selects 2r gloves where 2r < n. In terms of n and r, what is the probability that there is no matching pair in the sample? 4) A bowl contains r red balls and g green balls. One ball is select at random from the bowl and its colour noted, and it is returned to the bowl along with n additional balls of the same colour. Another ball is selected from the bowl (now with r + g + n balls) and it is observed to be green. Show that the conditional probability that the rst ball r selected was red is given by r+g+n . 5) Eight tires of dierent brands are ranked from 1 to 8 (best to worst) according to mileage performance. Four of the tires are chosen randomly. (a) Find the probability that the best of the tires selected was ranked 3rd. (b) Find the probability that the best tire selected was ranked 3rd and the worst 7th. (c) Denote the range as R where R = largest rank smallest rank. So in (b) R = 7 3 = 4. Find P (R = 4). (d) Give all possible values for R and the probabilities associated with each of these values. 6) Suppose the probability of exposure to the u during an epidemic is 0.6. Experience has shown that a serum is 80% eective in preventing an inoculated individual from getting the u if exposed. A person who has not been inoculated has a probability of 0.9 of acquiring it if exposed. Two people, one inoculated and the other not, perform a highly specialized task in a business. Assume that they are not at the same location, are not in contact with the same people, and cannot expose each other to the u. What is the probability that at least one of them will get the u? 7) A spinner can land in any of four positions, A, B, C and D with equal probability. The spinner is used twice and the position noted each time. Let the random variable Y denote the number of positions on which the spinner did not land. Compute the probabilities for each value of Y . STAT 2655 8) Five balls, numbered 1, 2, 3, 4 and 5 are placed in an urn. Two balls are randomly selected from the ve and their numbers recorded. Find the probability distribution for the following: (a) The largest of the two sampled numbers. (b) The sum of the two sampled numbers. 9) A rental agency, which leases heavy machinery by the day, has found that one expensive piece of equipment is leased, on average, only one day in ve. If rental on one day is independent of rental on any other day, nd the probability distribution of Y , the number of days between a pair of rentals. 10) Let Y be a random variable with P (Y = y) given by: y 1 2 3 4 . P(Y=y) 0.4 0.3 Find E[Y ], E[1/Y ], E[Y 2 1] and V [Y ]. 0.2 0.1 Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner